| L(s) = 1 | − 2·4-s + 4·9-s + 3·16-s + 8·19-s − 8·25-s − 8·36-s + 8·49-s − 48·59-s − 4·64-s − 16·76-s − 6·81-s + 48·89-s + 16·100-s + 24·101-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 32·171-s + ⋯ |
| L(s) = 1 | − 4-s + 4/3·9-s + 3/4·16-s + 1.83·19-s − 8/5·25-s − 4/3·36-s + 8/7·49-s − 6.24·59-s − 1/2·64-s − 1.83·76-s − 2/3·81-s + 5.08·89-s + 8/5·100-s + 2.38·101-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-s + 2.44·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182230878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182230878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.418839338644338202652010668407, −8.967506047524813963148122237658, −8.956520327368728494365539279328, −8.860554645954631556466625401263, −8.231272092882065147406641620987, −7.77265927397674281762515943603, −7.65822435266637210201028614554, −7.56179581696777197603083861310, −7.45775701579912025621558633254, −6.95699127394199851292202224631, −6.41504184010593481425340639787, −6.23266659380700618461353091116, −5.99433283283112592830123563923, −5.66876859062903956842955748051, −5.10054656720727117602642902702, −5.06086884440264440471390678977, −4.58024683313541212341029973976, −4.28697831313215297950563479453, −4.14911131052640219693930101612, −3.40772590844707343908426910388, −3.31581171376736480719605291617, −2.97024208037910435614385931099, −1.91505430413316352158022021370, −1.76197003093654825177348797755, −0.850154944955691292142113475389,
0.850154944955691292142113475389, 1.76197003093654825177348797755, 1.91505430413316352158022021370, 2.97024208037910435614385931099, 3.31581171376736480719605291617, 3.40772590844707343908426910388, 4.14911131052640219693930101612, 4.28697831313215297950563479453, 4.58024683313541212341029973976, 5.06086884440264440471390678977, 5.10054656720727117602642902702, 5.66876859062903956842955748051, 5.99433283283112592830123563923, 6.23266659380700618461353091116, 6.41504184010593481425340639787, 6.95699127394199851292202224631, 7.45775701579912025621558633254, 7.56179581696777197603083861310, 7.65822435266637210201028614554, 7.77265927397674281762515943603, 8.231272092882065147406641620987, 8.860554645954631556466625401263, 8.956520327368728494365539279328, 8.967506047524813963148122237658, 9.418839338644338202652010668407
